Fractal atomic surfaces of self-similar quasiperiodic tilings of the plane

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Fractal atomic surfaces of self-similar quasiperiodic tilings of the plane

C. Godrèche, J. Luck, A. Janner, T. Janssen

To cite this version:

C. Godrèche, J. Luck, A. Janner, T. Janssen. Fractal atomic surfaces of self-similar quasiperi- odic tilings of the plane. Journal de Physique I, EDP Sciences, 1993, 3 (9), pp.1921-1939.

�10.1051/jp1:1993221�. �jpa-00246839�

Classification Physics Abstracts 61.40 61.50E

fhactal atondc surfaces of self-sindlar quasiperiodic tilings of the

plane

C. Godrbche

(~),

J. M. Luck

(2),

A. Janner (~) ^{and T. Janssen}

(~)

(~) ^{Service de} Physique de l'itat Condensd, Centre d'itudes _de Saclay, 91191 Gif-sur-Yvette

Cedex, France

(~) ^{Service de} Physique Thdorique, Centre

d'itudes

_{de Saclay, 91191} Gif-sur-Yvette Cedex, France

(~) Institute for Theoretical Physics, University of Nijmegen, Toemooiveld 1, 6525 ED Nijmegen, The Netherlands

(Received

I February1993, accepted 19

May1993)

Rksum4. Nous considdrons _en para1l41e trois pavages quasipdriodiques auto-similaires du

plan, de sym4trie de rotation d'ordre huit, et constitu4s des deux mimes tuiles _: le carr4 et le losange £ 45 degrds. Les trois pavages ^sont ddcrits _par les mdmes rkgles d'inflation, I

une

permutation des tuiles prks. Nous dtudions l'influence de cette permutation _sur les surfaces

atomiques, _ou domaines d'acceptance, et _sur les spectres de Fourier des pavages. Le bord de la surface atomique de deux des pavages est fractal _pour l'un d'entre

eux ce bord n'est _pas

connexe. Les propridtds ddcrites _sur cette famille d'exemples sont vraisemblablement gdndriques.

Abstract We consider in parallel ^three self-similar quasiperiodic tilings of the plane with

eight-fold symmetry, made of two prototiles, the _square and the 45-degree rhomb. They _possess the same inflation rules up ^to a reordering of the tiles. We study the consequences of this

reordering _on the nature of the atomic surfaces, _or acceptance domains, and _on the Fourier spectra ^{of the} tilings. For two of the tilings the atomic surface has _a fractal boundary. For _one of them it is not _a connected set. We _argue that the situation described in this _paper is generic.

1 Introduction.

This _paper is an extension to the two-dimensional _case of _a recent

study

^devoted to the nature of the atomic surfaces of

quasiperiodic

substitutional

tilings

of the line

iii,

that is, structures

built with _a finite number of

prototiles (bonds

in _one

dimension),

and

generated by

so-called inflation rules _or substitution rules.

In _one dimension, it is _easy to invent _as many substitutional structures _{as one} wishes.

Furthermore,

for _a

given

substitution matrix

(counting

the number of tiles of each

species

after _one inflation step,

starting

from each

tile),

all

orderings

of the

prototiles

_are

possible.

Restricting

to

binary quasiperiodic

structures, characterized

by

substitution matrices with _a determinant

equal

to +I,

implying

the Pisot property

(the leading eigenvalue ii

> I, the other

one

being

such that (12( ^<

1),

it is well-known that _one may lift _up the

positions

of the vertices of the one-dimensional

tiling

into _a two-dimensional superspace. This has _two virtues.

Firstly

it is _a convenient form of book

keeping

^of the

positions

^of

vertices; secondly

it enables _{us to} reveal the

underlying periodic

order

existing

in superspace.

Indeed,

_a

quasiperiodic

structure can be obtained

by cutting

a

periodic

_array of bounded atomic surfaces

by

the

physical

_space

(a

line in the present

case)

_[2].

It _was shown in reference [I]

that, given

_a one-dimensional

quasiperiodic

substitutional structure, _one should expect ^{that the atomic surface}

describing

its

periodic

order in superspace has

generically

_a ^fractal

boundary.

For

instance,

if _one takes the Fibonacci substitution matrix

MF ₌

(l.I)

1 °

corresponding

to the rule A _-

AB,

B _-

A,

and squares it, _one gets

Ml

=

l~ ^(l.2)

corresponding

to six different substitutions

according

to the order of the letters A and B in the inflated words.

Among these,

four lead to structures with

regular

atomic

surfaces,

whereas the other two possess fractal atomic surfaces. Hence the fractal _nature of the atomic surface is coded in the

ordering

^{of the}

letters,

not in

spectral properties

of the substitution matrix.

We want to show that the _same ideas hold for substitutional

tilings

of the

plane, I-e-, tilings generated by

inflation rules _or substitution rules. Yet the task is far _more difficult. Whereas in

one dimension, _as said above, _{one may} invent _as many substitutional _{structures as one}

wishes,

this is _no

longer

true in two dimensions. An

arbitrary

substitution is not

implementable

for

tilings

with _a finite number of

prototiles,

due to the constraint of _space

filling. Furthermore, given

a substitutional

tiling

made of _a finite number of

prototiles,

it is not obvious that

changing

the

ordering

of the

prototiles

in the rules still leads to _an

edge-to-edge tiling.

The

study

made in this _paper is restricted to _one

family

of

examples,

that _we believe

generic.

We

investigatd

in

parallel

three

tilings.

The first _one is the well-known

eight-fold symmetric

Ammann

tiling,

made of _a rhomb and _{a square} [3]. The second _{one can} be found in reference [4].

The last _one is novel.

The three

tilings,

^denoted

by A,B,C

in the

following,

are

quasiperiodic. They

have the

same symmetry, are made of the _same two

tiles,

in the _same

proportions,

but with different

geometrical orderings.

We will show

that,

_as in the one-dimensional _case, these

changes

^of

ordering

lead to fractal atomic surfaces. We will also illustrate the consequences of this fact

on Fourier spectra. As noticed for the one-dimensional _case, the Fourier spectrum ^of a sub- stitutional structure with _a

regular

atomic surface is

"sharper"than

that of _{a structure} with

a fractal atomic surface. In other terms, the

fractality

of the atomic surfaces

implies

less _reg-

ularity

in the Fourier spectra, ^and

especially

_a slower fall-off of the

intensity

of

Bragg peaks.

Loosely speaking,

the _more fractal the

surface,

the _more the

peaks

_are visible in thd spectrum.

The

quantitative explanation

of the

phenomenon

_was

given

in reference [I] ^{for the} one-

dimensional _case. Here _our

approach

will be _more

descriptive.

The existence of

tilings

with fractal atomic surfaces has been known for _some time. One

example

_can be found in reference [5].

Recently

_a number of _new

examples

have

appeared

[6-8].

Nevertheless _none of these references considers in _a

systematic

_way the

question

under

study here, namely

the consequences of the

change

of the

ordering

of the tiles in the inflation rules.

This _paper is

organized

_as follows. Section 2 is devoted to the

description

of the three

tilings

defined

by

inflation rules in

physical

_space. In section 3 _we discuss the nature of the

corresponding

atomic surfaces. In _{two cases} their boundaries _are fractal

objects,

for which _we

give explicit

construction rules and the

analytical expressions

of their dimensions. In section 4

we show

graphs

^{of the} Fourier spectra ^{of each}

tiling, computed by

_means of recursion relations between Fourier

amplitudes,

deduced from the inflation rules

defining

the

tilings.

An

appendix gives

some

insight

_on the _way of

defining systematically

_a

family

of

tilings possessing

the _same

substitution

matrix,

at least for the _case under

study.

2. Definition of the

tilings.

In this section _we describe three self-similar

tilings,

the inflation rules of which share the _same substitution matrix.

They only

differ

by

the

ordering

of the tiles in the

patch

of tiles obtained after _one inflation step. ^{We will} see that this fact has

far-reaching

consequences, as alluded to in the introduction.

The first _one is the well-known Ammann _or

octagonal tiling,

hereafter denoted the

A-tiling.

This

tiling

_may be built

by

inflation rules [3,

4,

9, 10,

11],

or

by cut-and-projection

from _a 4-dimensional _superspace [10, 12]. ^{It also} possesses

matching

rules [3,

10].

The second

tiling,

hereafter denoted the

B-tiling,

is described in reference [4]. The third one, hereafter denoted the

C-tiling,

is novel.

These

tilings

_are made of two

prototiles,

the

45-degree

rhomb and the _square, with

equal

sides. It is _more convenient to describe their construction in terms of the

rhomb,

denoted

by

R, and the

45-degree

isosceles

triangle,

denoted

by

T, obtained

by cutting

the _square

along

one of its

diagonals.

Their inflation _-or substitution- rules _are shown in

figures

la-c

(conventions

used in these

figures

will be

given below). Leaving

aside _any

geometrical description,

these read

R _- 3R + 2T + 2T'

aA T _- 2R ₊ 2T + T'

(2. la)

T'- 2R + T + 2T' R-R+2R'+4T

aB R'

- R' ₊ 2R ₊ 4T

(2. lb)

T-R+R'+3T

R _- 2R + R'+ 4T

R'- R ₊

2R'+

4T'

ac

(2.lc)

T _- R + R' ₊ 2T ₊ T'

T'

- R + R' _{+ T +} 2T'

R' and T' _are obtained from R and T

by

_a mirror symmetry -or reflection- in the

plane.

In the

Appendix

_we explain how these rules _may be derived in _a

systematic

_way.

T

~ T

~ T'

R R

T

~

T R

~ ~ ~. ^R

~

a)

T T

R' R R'

T ~

~ T T

~ ~. ~,

~

R _{T T}

~

R' R

~ ~

T T T T T T R R

e~ ~

b) ^c)

Fig. 1. Inflation rules for the A-, B- and C-tilings.

With

regard

to combinatorial aspects it is not necessary ^to

distinguish

between the

prototiles

and their mirror

images.

^{Hence all three}

tilings

_are described

by

the _same substitution

R _- 3R ₊ 4T

j2.2)

° T _{_} 2 R + 3T

The associated substitution matrix,

counting

the number of tiles of each species obtained after

one inflation step,

starting

^{from either} R _or

T,

reads

M =

(~ ()

=

(~ ~) ^12.3)

~

Its characteristic

polynomial

is

P(1)

⁼¹²

61+1,

_so that its

eigenvalues

_are

&=3+24=@), @1=1+V5

12=3-24=@(, @2"1-v5

^~~~~

Since the matrix M has _a unit determinant, and

obeys

^therefore the Pisot property, the

tilings

under

study

_are

quasiperiodic (see

Sect.

4).

Aiter each step ^oi inflation, _a

patch

oi tiles is

enlarged by

_a linear factor _@I The normalized

right

Perron-Frobenius

eigenvector

associated with the

eigenvalue

ii

yields

the frequencies ^{of each}

prototile

in the three

tilings,

namely

pR

=

v5

_1,

_p~

= 2

V5 (2.5)

whereas the components of the left

eigenvector

_are

proportional

to the _areas of both

prototiles,

which read

A~

"

~l' ^A~

"

(2'6)

in units where the rhomb and the square have sides

equal

to I. Since

ARp~

=

A~p~,

the _areas covered

by

rhombs and

triangles

_are

equal

in the infinite

tilings.

e~

e4 e~

ej

Fig. 2. Unit vectors in the plane of the tilings.

In order to describe

accurately

the

geometrical

content of the inflation rules

generating

the infinite

tilings,

let _us introduce the

elementary geometrical

operations that build the inflated tiles from the

prototiles

_[13]. We denote

by

R the rotation of

angle x/4

around _a chosen

origin

of the

plane

of the

tiling

and

by

_ei>

, e4 unit vectors in the

plane (see Fig. 2).

Let _us focus

on the Ammann

tiling

in order to illustrate _{our purpose.} In _terms of these

operations

the inflation rules _are

Rn+i

"

[R°,

0] Rn +

[R~, (ei

+ e2

e4)

_@/] Rn

+

[R~, (el

+ _e2 + _e3

e4)

_~~j

Rn

+

[R°,

_{ei @/]} Tn +

[R~, (ei

+ 2e2 + e3

e4)

_@/] Tn

+

[R~~, (ei

_{+ e2 +}

e3)

_@/]

T[

₊

[R, (ei

_{+ e2}

e4)

_@/]

T[ (2.7a)

Tn+i

_"

[R, ei)

Rn +

[R~~, (ei

+

e2)

_@/] Rn

+

[R~~, (ei

+

e3)

_@/]

Tn

₊

[R~, (2ei

+

e2)

_@/]

Tn

+

[R°,

_{ei @/]}

T[ (2.7b)

T[

~i ⁼

[R°,

_0]

Rn

+

[R~, (ei

+ _e2

e4)

_@/]

Rn

+

[R°,

_{ei @/]} Tn ₊

[R~~, (ei

+ _e2 +

e3)

_@/]

T[

+

[R~, (2ei

+ e2

e4)

_@/]

T[ (2.7c)

Rn,

Tn,

T[

denote the

patch

of tiles obtained at the n-th iteration step,

starting

from Ro

=

R,

To

"

T,T[

= T'. These

patches

_are inflated tiles. In other terms the inflation rules _are

equivalent,

_up to _a

rescaling,

to

cutting

rules _-or tessellations- of the

prototiles. By

convention, all tiles

(prototiles

_or inflated

tiles)

have their

origin

at their left _corner indicated

by

_a dot in

figures

la-c. Their orientation is

given

in

figures

la-c.

By

convention T' _{is drawn}

as

T,

I-e-, with its basis

horizontal,

and its

origin

on the left vertex. Arrows allow _us to

distinguish

between _a tile and its mirror

image (see

also the

Appendix).

In

equations (2.7)

the notation

[.,.]

^{should be understood} as a rotation

acting

_on the

patch first,

^followed

by

_a translation.

These

equations provide

_a

practical

_means of

constructing

the

tiling. They

will also allow _us to compute its Fourier spectrum

(see

Sect.

4). They

_may be

equally

understood _as

describing

rotations and translations in the 4-dimensional superspace. Similar

equations

_may be written for the other _two

tilings.

a)

b) c)

Fig. 3. Patches of tiles obtained after 4 steps of inflation for the three _cases, starting from _a unit square.

Figures

^{3a-c show the}

^patches

^{of tiles obtained}

by iterating

the rules

(2.I)

four times,

starting

from _a unit square, I-e-,

by juxtaposing

the

triangles

T and T'

along

their

basis,

for the three

tilings.

^Each

patch

consists df 816 rhombs and l154

triangles,

in agreement ^with

(l154) ^~~ (2)

~~'~~

Triangles

_come in pairs and build squares, as

expected.

To

summarize,

the three

tilings

shown in

figures

3a-c share the _same substitution matrix

M, given

in

equation (2.3),

if _one does not discriminate between _a tile and its mirror

image. Hence,

at each

given generation,

the three inflated

patches

contain the _same numbers of rhombs and squares. Nevertheless the three

tilings

described in this section have _a number of

distinguishing

features. For instance, _a

simple

inspection of

figures

3a-c shows that _new

"vertex-stars"appear

in the

B-tiling

that did _not exist in the

A-tiling.

The _same is true for the

C-tiling.

For

example, configurations

with four _squares incident at _a vertex may be found in the

B-tiling,

and

configurations

with three _squares and _two rhombs incident _{at a} vertex in the

C-tiling.

A

systematic

_way of

studying

these different vertex-star environments consists in

making

_use of the _superspace

description

of the

tilings given by

their atomic surfaces. Another

distinguishing

feature is their Fourier spectra. These two

points

will be the

subject

of the next two sections.

3.

Superspace representation

and atomic surfaces.

The four unit vectors ei, e2, e3> and _e4, shown in

figure

2,

together

with their

opposites

_are,

up ^to a

scale,

the

orthogonal projections

onto _a two-dimensional

plane,

the

physical

_space

E",

of _an orthonormal basis

(e),e(, e(,e()

of the four-dimensional Euclidean superspace. Since the

tilings

consist of _squares and

45-degree rhombs,

which

only

_assume

eight

orientations, ^all

their vertices _can be reached

by moving

steps

along

_one of the

eight

unit _vectors shown in

figure

2,

starting

from _an

arbitrary origin.

^The

tilings

_can thus be lifted in superspace, the vertices

being mapped

onto

points

with

integer co-ordinates,

I-e-

points

of the lattice Z~. The

internal,

or

perpendicular,

_space

E~,

_is defined _as the two-dimensional

plane orthogonal

to

E".

_The

projections

of the basis vectors of _{superspace onto} E~

are, up ^to a

scale, given by

the four unit vectors

et, e), e),

and

et,

shown in

figure 4, together

with their

opposites.

e~1 e41

e~1

Fig. 4. Unit vectors in internal space E~

To summarize, an

arbitrary

vertex of either

tiling, represented by

the vector XII _in

physical

space

E",

_is in one-to-one

correspondence

with _a lattice _vector

X~

_in

Z~,

with four

integer

co-ordinates

jai,

_{n2> n3,}

n4),

and with _a vector X~ _{in internal}

space

E~,

_so that _we have

4 4 4

XII

=

£

_nkek>

X~

=

£ nke(, X~

=

£ nke) _(3.1)

k=I k=I k=1

The inflation

operation

associated with the rules

(2.I)

is

represented

in superspace

by

the

following integer

matrix

1 1 0 -1

~~~

_~~'~~

-l

0

~

which admits two two-dimensional

eigenspaces, namely ^E"

and

E~,

the

corresponding eigen-

values

being

_@i and _@2

respectively.

We

define,

_as in reference

[I],

the atomic surface of _a

given tiling

_as the closure of the countable _set

(X~

of the vertex

positions

of the infinite structure in internal _space.

In the case of the

A-tiling,

the

projection algorithm

allows _one to determine the atomic surface in _an exact fashion

[10, 12].

^{It is} a

regular

octagon with _a unit

side,

centered at the

origin,

denoted Q. The atomic surface has therefore _{an area} (Q( =

2(1+ vi).

The B

-tiling

does _not admit _any

simple

construction

by projection.

We _can therefore expect that its atomic surface _{is a} richer

object. Using

_a

straightforward pointwise

construction of the atomic

surface,

_we have met with the

following empirical

fact. In order to get domains of internal space which _are filled

uniformly by

the vertex

positions X~,

_one has to separate the vertices into two

equally populated

_sets,

according

to the

parity

^of the _sum of their _superspace

co-ordinates

(n~).

We _are thus led _to introduce the

following

two sets

Seven "

(X~

_{al + R2 + R3 + R4}

_even)

Sodd ₌

(X~

_{ni + n2 + n3 + n4}

odd)

^{~~ ~~}

where the overline denotes the closure of the _set.

Figure

5 shows the atomic surface Seven. We _can make the

following

observations. Sodd is obtained from Seven

by

_a

45-degree-rotation

around the

origin.

Seven is contained in the square built _on the _even A points

(A2>A4, A6,

_A3)> whereas Sodd is contained in the _square built _on the odd A points

(Ai

A3> A5>

A7),

with the notations of

figure

6.

Finally,

the

boundary

of the

atomic surfaces looks fractal.

Let _us recall that the inflation

operation

acts in E~ _{as the}

multiplication by

_@2

" 1-

v5,

I-e- _{as an}

isotropic

contraction. In

analogy

with the _case of

chains,

it _can be shown that the atomic surfaces Seven and Sodd _can be partitioned into

finitely

_many subsets, which _are similar among

themselves,

with _{a common}

similarity

ratio of _@2.

Instead of

working

out this cumbersome program

fully,

_we

adopted

_{a more} heuristic ap-

proach.

We notice first that the above mentioned

scaling

ratio shows _up in the construction of

figure

6 in the

following simple

_way. We have _e-g- (A4B3( =

Vi12,

_{and (A4B2(}

" 1 +

v5/2,

so that the ratio

(A4B3(/(A4B2(

is

equal

to (@2(. The natural

assumption

that the _arc

(A4B3)

of the

boundary

^of

SE

is similar _to the

larger

_arc

(A4B3C3B2),

with the ratio (@2(,

together

with the

global

symmetry ^{of the atomic}

surface,

allows _us to construct its

boundary

_{as a} closed self-similar fractal _curve, which _we show in

figure

7.

Although

_we have _no

rigorous proof

that

this _curve is

exactly

the

boundary

of the atomic

surface,

_{we are} convinced that this is the _case.

Indeed, _we found a

perfect

_agreement between the pointwise construction of

figure

5 and the

,"~~',

B~ ," ",

B~

~4 ', ,'

,

~z

", _,"

',,'

B~ ^' ~3

B~

,,~ ^{~4 ~z} _{~~. ,~~}

j)ij )j

j C~ Cj

("

_'~'

""

""

C~ C~

"" "'

~~~~~j

C~ T~~~~

,,',

,, ',

A~ ^' ,,' ",

A~

Fig.

5

Fig.

6

Fig. 5. Atomic surface Seven of the B-tiling obtained by pointwise projection onto internal _space of the patch of tiles obtained after _{n =} 6 inflation steps.

Fig. 6. Geometrical framework of construction of the boundary of the atomic surfaces for the

B-tiling-

fractal _curve of

figure

7.

Moreover, constructing

_a fractal _curve with the

required properties (such

_{as area (Q(,}

similarity

ratio @2) is a very constrained

problem.

The dimension dB of this _curve reads

dB ₌

~(

= l.246477

(3.4)

n i

The atomic surface Seven thus consists of the octagon

Q,

which is the atomic surface of the

A-tiling, plus

four

triangular-shaped regions,

such _as

(BiA2B2)>

minus four

triangular-shaped regions,

such _as

(B2C3B3)

Since the

eight triangular-shaped regions

are

isometric,

_we obtain in

particular

that both

partial

atomic surfaces Seven and Sodd ^have the _{same area as} the octagon Q.

The situation for the

C-tiling

^is similar to that encountered for the

B-tiling, though

of _a

higher

level of

complexity. Again,

_one has to introduce two atomic surfaces Seven and

Sodd,

transformed into each other

by

_a

45-degree-rotation

around the

origin. Figure

8 shows _a

plot

of Seven. The rules used to build the fractal

boundary

of the atomic surfaces have

again

been elaborated from

empirical

observation. The exactness of this construction is nevertheless

beyond

_any

doubt,

for the _{same reasons as} above.

The construction rules involve three types ^of oriented segments

joining

black and white vertices, _as shown in

figure

9. The matrix associated with these rules is

0

⁰

0 0

(3.5)

3 6 2

from which _we deduce that

dc ₌

~~~(~/~~

= l.618000

(3.6)

n 1

4_y,

J '~.

~i.

,~ (~$~

J J

<

J

J

/

~

^fl

I', ':

i'i «d~

~

Fig.

7

Fii.

₈

Fig. 7. Boundary of the atomic surface Seven for the B-tiling, obtained by applying the rules of

figure 6 ad infinitum.

Fig. 8. Atomic surface Seven of the C-tiling obtained by pointwise projection onto internal _space of the patch of tiles obtained after _n

= 6 inflation steps.

2

/~

3

.-- ~

§_

.---

~

₃

~ fi

,

3

/~$

.-- ~ .---

Fig. 9. Rules of construction of the boundary of the atomic surface for the C-tiling.

The recursive construction rules shown in

figure

9 have to be

applied

to the octagon ^{Q. With} the notations of

figure

6, the atomic surface Seven

corresponds

to

taking

for initial condition the vertices Bk of the octagon, ^labelled

by

white dots when k is

odd,

and

by

black dots when

k is even. Black and white dots _are

interchanged

in order _to generate ^the atomic surface Sodd.

Figure

10 shows the

boundary

of Seven. It consists of _an

infinity

of

disjoint

fractal closed curves, so that the atomic surface is made of _many

infinitely

connected components, ^which

contain

infinitely

_many holes.

Every single

closed _curve has _a dimension

dB,

smaller than the dimension

dc

of the

boundary

_{as a} whole. The _area of either atomic surface coincides with that of the octagon

Q,

_on which the recursive construction is based.

,v, ,v,

~ ^~~$~

'~i~

~~_

?'

~$~~$

.

't j'

>:

~~~ ^~~~

t«

~if~

~i~

~ jj~.

$$f~j/

~

~fl~

.~» ,~.

Fig. 10. Boundary of the atomic surface Seven for the C-tiling, obtained by applying the rules ol'

figures 9 ad infinitum.

The diameters of the first _two atomic surfaces read AA

"

(BiB5(

"

~fi,

_{and AB}

"

(A2A6(

_" 2 +

v5.

_{In the}

case of the

C-tiling,

^{the diameter} reads Ac

"

2(OEI

_Ii where Ei is

one of the

eight

extremal

points

of the atomic surface Seven. Its

co-ordinates, namely

Ei ^~

~/~,

^~

~/~l, (3.7)

have been obtained

by identifying

the fixed

point

of _a

carefully

chosen fourth iterate of the rules of

figure

9. We thus have the

following

numerical values

A

tiling

dA

= I, AA

" 2.613126

B

tiling

dB ₌

1.246477, AB

_" 3A14214

(3.8)

C

tiling dc

= 1.618000,

Ac

₌ 4.249530

which show that there is _a

good

correlation between the diameters of the atomic surfaces and the dimensions of their boundaries.

As

already

mentioned, the three

tilings

considered in the present ^work are

quasiperiodic

structures. The associated atomic surfaces of the B and

C-tilings

_are

bounded,

in spite of the fractal nature of their

boundary. They

have _a finite _area,

namely

that of the octagon

Q;

their fractal

boundary

_can thus be viewed _{as a} decoration of the octagon. As _a consequence, the

frequency

of _any

configuration c",

_{like e-g-} a "vertex

star",

_can be evaluated via the usual rule,

namely,

it is

proportional

to the _area swept in

translating

the

image c~

of the

configuration

in

E~,

under the constraint that this image is included in the atomic surface.

InURNAL DE PHYSIQUE -T i N.~ SEPTEMBER (ml ?<1

Moreover, the

periodic

_arrays of atomic surfaces of the three

tilings

leave _no holes in the internal _space

E~

_{This property is clear}

by

construction for the

A-tiling

_[10]. It is less obvious in the other _{two cases,} arid

especially

for the

C-tiling,

since it

implies

that all the holes of _one atomic surface Seven _are

exactly

filled

by

the disconnected "islands" of _an immediate

neighbour- ing

^surface

^Sodd,

^and vice-versa. This characteristic is shared

by

all substitutional _sequences

[I].

As _a consequence, if the

physical

_space is deformed in _a smooth way in superspace with respect to the linear space

E",

_{the atoms rearrange} themselves

locally,

without

disappearing

or

hopping

to

large

distances. In _more

physical words,

the

phasonic degrees

of freedom have _a basis made of

elementary

local _{moves, or}

"flips".

4. Fourier spectra»

As _seen in the

previous section,

the three

tilings

under

study

differ

by

their atomic surfaces.

We will _now show that this is reflected in the behaviour of their Fourier spectra. ^{As recalled} in the introduction, it is indeed

expected, by analogy

with the one-dimensional _case

[I],

that the

larger

^the fractal dimension of the

boundary

of the atomic

surface,

the less ordered is the

tiling,

hence the _more

complex

is its Fourier spectrum.

Let us denote

by fin _(q)

_and

in _(q)

_{the Fourier}

_amplitudes

_{associated to} the

patches

of tiles Rn _or

Tn,

I-e-, the Fourier transforms of the

density

of matter

Rn(x)

and of

Tn(x)

set _on the

tiles

(and

similar notation for the reflected

patches

R' and

T').

We will compute ^Fourier spectra

by using

recursion relations between Fourier

amplitudes

deduced from

equations (2.7) (and analogous equations

for the other two

tilings)

_[13]. For the

case of the Ammann

tiling they

read

j~n+1(~)

"

~n(~)

₊

~XP[~~(~' (~l

+ _e2

e4) ~/)j ~n(~ _~~)

+

eXp(-I(q (el

_{+ e2 + e3}

e4) ~/j ~n(~)

+

eXpj-iq'el

_@?I

inlq)

+

exp[-iq (ei

₊ 2e2 _{+ e3}

e4)

_@/]

in (R~~q)

+

exp[-iq (ei

+ e2 +

e3)

_@/]

I[(R~q)

+

~XPl~iq l~l

_{+ ~2}

_~4)

_@?I

illR~~q) _14'l~)

tn+llq)

"

eXpj-ijq

_{el @t)I}

llnlR~~q)

+

eXpj-ijq'jel

₊

~2)

_@?I

llnlRq)

+

eXp[-lq' (el

₊

e3) ~/j ~n(~~~)

+

exp[-iq (2ei

+

e2)

_@/]

in(R~~q)

+

exp[-iq

_{ei @/]}

I[ _{(q) (4. lb)}

fl+I(q)

"

lln(q)

+

exPl-I(q (ei

+ e2

e4)

_@/1

llnlR~~q)

+

expj-iq

_{ei @/I}

inlq)

+

exp[-iq (ei

_{+ e2 +}

e3)

_@/]

I[(R~q)

+

eXpj-iq' (2el

+ _~2

~4)

_@?I

il(R~~~) _(4'l~)

These relations should be

complemented by

initial conditions which

depend

_on the choice of the

density

of matter _on the tiles. We choose to put _a

pointlike

atom of unit _{mass on}

each vertex of the

tiling. Consequently,

each vertex of _a tile bears _a delta function with _an

amplitude proportional

to the inner

angle

at this vertex.

Figures

lla-c show the Fourier spectra

a)

b)

Fig. ii. Fourier spectra corresponding to the three patches of figures 3a-c

(see text).

C) Fig. ii.

(continued)

corresponding

to the three

patches

of tiles of

figures

3a-c obtained after 4 steps of

inflation, starting

from _a unit _square, with -27 < q~,qy ^< 27. In order to avoid four-fold

symmetric

finite-size effects

generated by

the outer

shape

of the unit square, we have

actually computed,

for each

tiling,

the Fourier spectrum ^of a coherent

superposition

of

eight

copies ^of the

patches,

obtained _one from the other

by

successive rotations of 45

degrees.

The

graphs

_are made of 512 _x 512

pixels.

Some

intensity

would be lost for _a number of steps > 5.

Indeed,

the resolution

reads

2x/&q

_m 59.6 _m @).6~ This representation is the best

compromise

we have found.

The _common Fourier module of the three

tilings

_can be determined from the recursion

relations

(2.7) (and

similar

Eqs.

for the B- and

C-tilings) [13-16]. Bragg peaks

are indexed

as

4

q = 7r

£

_{nk ek}

_(4.2)

k=1

where _{nk are}

integers.

A

progressive

increase in

complexity

is observed _on the Fourier spectra ^when

going

from the

A-tiling

to the

C-tiling,

in agreement with the considerations of

previous

section. As mentioned in the

introduction,

this statement could be made _more

quantitative by

_means of _a

study

of the fall-off of the intensities of

Bragg peaks.

In the one-dimensional _case

[I],

the _power law

governing

this fall-off _was shown to be related to the fractal dimension of the

boundary

of the atomic surface. In the present case of

tilings, figures

lla-c show that the number of

visible

Bragg

diffractions is

increasing

with the dimension of the

boundary.

A

quantitati,,e study

of this

phenomenon

^is made _more difficult

by

the presence of four

integers

in equation

(4.2),

instead of _one in the _case of

binary

chains.

5. Conclusion.

Let _us put ^the main

points

of this _paper in _{a more}

general perspective.

One _may first ask the question of the amount of

fractality

that _may be

expected

_a priori in the

geometrical

features of self-similar

tilings.

In

particular

_{one may} wonder to what _extent

atomic surfaces with _a fractal

boundary

_are "natural".

. Self-similar

objects

_are

generically fractal,

I-e-,

they

have _a

non-integer

dimension d and

they

_are non-smooth down to

arbitrarily

small scales. But

they

_can be in _{some cases} very

regular objects.

The patterns

generated by

the so-called iterated

fltnction

systems

(see

_e.g.

Ref.

[17]),

as well _as the Julia sets of

holomorphic dynamical

systems,

provide

illustrations of this statement. In the latter

example (see

_e.g. Ref.

[18]),

^it is known that Julia sets are

"very fractal",

unless

they

_{are very}

simple, namely

_a

straight

line segment or a circle.

. Atomic surfaces cannot

belong

to that

general

class of fractal

objects,

because their

dimension,

which is that of

E~,

_{is an}

_{integer (n}

= 2 in this

paper),

and their volume

(area

for _n

=

2)

is

finite.

They

are nevertheless "as fractal _as

they

_can

be", taking

this constraint into account,

namely, they

exhibit fractal

boundaries,

either in the _case of _{sequences, or} in the present

situation of two-dimensional

tilings.

. Substitutional

tilings

cannot exhibit _any

simple

kind of

fractality

in real space,

recognisable

at first

glance,

since

they

must fulfill the constraint of

being

space

filling. Nevertheless,

the richness of their Fourier spectra suggests that the

complexity

which goes hand in hand with

self-similarity

is still present. It is also worthwhile

recalling

that non-Pisot substitutional structures, which have _no

Bragg peaks

in their diffraction spectra, do exhibit

fractality

in real space, in the _very

expression

of the unbounded fluctuations of the atomic abscissas with respect

to the

underlying

_average lattice [19,

20].

We _{now come} back to the classification of

tilings generated by

substitutions in order to discuss the

generality

of the results

presented

in this paper.

Assuming

that _~i,e know how to build the infinite set of such

tilings,

if _one

picks

"at random" _one member of the _set

(hence

the

corresponding

substitution

matrix),

then with

probability

_{one a} non-Pisot

tiling

will be found.

This relies _on the fact that _a substitution _may be characterized

by

its Perron root, ^{which is}

an

algebraic integer,

and that non-Pisot numbers _are

overwhelmingly

_more

frequent

amongst

algebraic integers

than Pisot numbers.

Then, restricting

to Pisot substitutional

tilings,

_one

has first to discriminate between substitutions with _a determinant equal to + I from those with _a

larger

determinant

(in

absolute

value).

The first category

corresponds

to

quasiperiodic tilings,

the second _one to

limit-periodic

_or

limit-quasiperiodic tilings

[1, 15, 16].

Restricting

to

quasiperiodic tilings,

_{one may} then discriminate between

tilings

with

regular

atomic surfaces and

tilings

with fractal atomic surfaces. Our

experience

of the one-dimensional _case leads

us to think that the latter _case is

generic.

To summarize,

tilings generated by

inflation _are

by definition,

self-similar. A

minority

of them is formed

by

Pisot structures, among which a

minority

is

quasiperiodic. Among

this last category a

minority

has

regular

atomic surfaces.

In

conclusion,

_a

tiling

in the _same time

quasiperiodic,

self-similar and with

regular

atomic surface is _a

rarity.

One is

tempted

to think that Penrose _was

lucky

to find _a

tiling

of this

kind,

since he used _an inflation method _to generate it. Nevertheless the chance to do _so is

larger

for inflation rules

involving

fewer tiles. Indeed the

genericity

of self-similar

tilings

described above becomes

fully

relevant when the number of tiles involved increases. The _same holds true for one,dimensional structures

[I].

For instance the Fibonacci structure has _a

regular

atomic

surface. Fractal atomic surfaces _appear for substitution matrices with

larger

entries.

Finally,

it is

interesting

to compare the infinite set of

tilings generated by

the inflation

method,

described above, to that

generated by

the

cut-and-projection

method

(or

section

method).

Indeed _{one may say,} to the risk of

simplifying

the

issue,

that

they

_are the two

main

global

methods

allowing

one to construct

tilings.

^{We thus exclude} _any consideration _on

matching

rules. In

particular

_{one may} wonder "how

large"

is the intersection of the two infinite sets of

tilings. Tilings generated by

canonical section methods _are,

by definition, quasiperiodic

and the associated atomic surfaces _are

simple. Among

them _a

minority

is self-similar. The intersection between the sets of

tilings generated by

the two methods is small. It consists of

tilings

which represent ^the

minority

of each set.

Hence,

the substitution method and the

section method

pull

structures in

opposite

directions.

Acknowledgements.

We wish to thank M. Baake for

informing

_us ^{of the existence of references} [6-8] ^{and for his}

comments on these

works,

_as well _as L.

Danzer,

M.

Duneau,

K. P.

Nischke,

and Z. Y. Wen for

interesting

discussions. The Dutch Foundation for Fundamental Research of Matter

(FOM)

is

acknowledged

for

partial

financial support ^{for the four of} us.

Appendix.

In this

appendix

_we

explain

how to

build,

in _a

systematic

_way, self-similar

edge-to-edge tilings

with the _same substitution matrix _as that of the Ammann

tiling,

called the

A-tiling

in the text. In other terms these

tilings

resemble the Ammann

tiling

_up to _a

reordering

of the tiles in the

patches

obtained after _one inflation step,

starting

from the

prototiles (the

rhomb and the

triangle).

The outer contours of these

patches

_are

again

_a rhomb and _a

triangle,

^homothetic to the

corresponding prototiles

with _a linear ratio of

@1 ^" 1+

vi.

All

possible orderings

^of

prototiles

inside these two contours are

given by

all

possible

tessellations of these inflated tiles. The various

possibilities

_are shown in

figures

12a-b. The

key question

is whether these tessellations

can be iterated. If _{so, a} self-similar

tiling

is

produced

and the tessellation

provides

_a substitution rule. If not, after _a few steps of

tessellation, configurations

_appear with tiles which _are not

edge-to-edge.

Let _{us now come} back in greater ^detail to each step of the reasoning.

(I)

The various tessellations _are in one-to-one

correspondence

with arrowed

prototiles

de- noted

RI

R4>

Ti

T5

(Fig. 13).

We will also denote

by

_an accent the

prototiles

obtained

by reflecting

the former _ones. The

correspondence

between the system of _arrows and the tes-

sellations is _as follows. All sides of

length

_@i of the scaled tiles of

figures

12a-b _are made of two

parts: an

edge

of _a

prototile

of

length

I and _an

edge

of _a prototile of

length v5 (the diagonal

of the _square, I.e. the basis of the

triangle).

The _arrow is drawn from the vertex contained

by

the

edge

of unit

length

towards the other _vertex. The _arrows borne

by

the basis of the

triangles

have _a similar

meaning.

(ii)

Next _we have to test the

self-consistency

of these

markings.

We illustrate the

reasoning

on one

example. Suppose

_we want to inflate Ri Let _us denote

by

_a the candidate inflation

given by

the tessellation. The inflated tile

«(RI)

should be

given by

the tessellation of

figure

12a but _now with _arrows marked _on the

edges

of Ri As _a consequence the _{arrows on} the

three

prototiles

Ri force _{arrows on} the four

triangles

contained in

a(Ri).

It is easy ^to see that the

only

arrowed

proto-triangles

which fulfill this system of _{arrows are} Ti or T4

land

their mirror

images). Looking

now at

a(Ti)>

_one discovers that

arrowing

first the two rhombs _as Ri is

compatible

with the

arrowing

^of the

triangles

_as Ti or its mirror

image T(.

The _same

reasoning

^shows that

arrowing

first the two rhombs _as Ri in

a(T4)

is not

compatible

with the

arrowing

of the

triangles

_as T4 or its mirror

image T(.

Other arrowed

proto-triangles

would be

coiupatible -namely

T~ _or T5- but then the candidate inflation rule would involve _more than